Galois group of $\mathbb{C}(t)/\mathbb{C}(t^{20})$

Question

Let $t$ be a variable. Can I say that $\mathbb{C}(t)$ is the splitting field of the separable polynomial $f(x)=x^{20}-t^{20}\in\mathbb{C}(t^{20})$, for then the Galois group is cyclic of order $20$? More specifically, the roots of $f(x)$ should be $t,\zeta_{20}t,\ldots,\zeta_{20}^{19}t$ where $\zeta_{20}$ is some primitive $20^\text{th}$ root of unity. But $\zeta_{20}$ is already in $\mathbb{C}$, so the splitting field is just $\mathbb{C}(t)$. The elements of the Galois group are just maps $\sigma_k:t\mapsto\zeta_{20}^kt$ for $0\leq k\leq 19$. Clearly $\sigma_1$ (or any $\sigma_r$ for $(r,20)=1$) is a generator for the Galois group, so it is cyclic on $20$ elements. Is this really all there is to it, or am I missing something?